Bearing estimation is an important topic in sonar and radar applications. Historically, this estimation has been performed by a process called beamforming. The conventional beamformer adds delays to the outputs of sensors along an array. The summation of these signals produces a beam steered in a direction determined by the delay intervals. The angular resolution is given by the wavelength to aperture ratio. This is known as the Rayleigh limit.
Many techniques have been used to increase the resolution of the direction-of-arrival problem. Minimum variance and eigenvector analysis are two techniques that exploit the structure of the sample covariance matrix. A promising technique is the maximum likelihood method. This is a robust method that works well even with low signal to noise ratios since it does not require inversion of the sample covariance matrix.
The maximum likelihood approach is a classical statistical method which fits the observed data to a parametric model. The best estimate occurs when a set of parameters are found that result in the minimum least squares error. This is also known as the maximum likelihood estimate of the parameters. Historically, the method can be traced back to Carl Friedrich Gauss. Currently, maximum likelihood, along with least squares minimization and chi-square minimization, are seen as the best methods to solve difficult nonlinear problems.
The maximum likelihood method is presently of considerable interest in solving signal processing problems. Theoretical calculations and simulations have shown that estimates based on this method statistically approach the Cramer-Rao limit. This limit is the optimum value derived from information theory. Unfortunately, the traditional maximum likelihood calculations required for this approach have been viewed as computationally too difficult. This is largely due to the complex matrix operations required to evaluate the likelihood function and the nonlinear multidimensional maximization process.
The traditional maximum likelihood method operates on a data vector that is squared to produce a sample covariance matrix. This data representation is used by other high resolution bearing estimation techniques including minimum variance and eigenvector approaches. A recent publication in the IEEE Signal Processing Letters, Vol. 1, number 12, pp 203-204, by Piper has shown how the complex maximum likelihood matrix operations can be reduced to a simple vector product. This maximum likelihood approach can be improved to account for an arbitrary number of sources and two dimensional (2-D) array geometries.